What is the concept of the technological singularity?
The technological singularity is a theoretical point in time at which a self-improving artificial general intelligence becomes able to understand and manipulate concepts outside of the human brain's range, that is, the moment when it can understand things humans, by biological design, can't.
The fuzziness about the singularity comes from the fact that, from the singularity onwards, history is effectively unpredictable. Humankind would be unable to predict any future events, or explain any present events, as science itself becomes incapable of describing machine-triggered events. Essentially, machines would think of us the same way we think of ants. Thus, we can make no predictions past the singularity. Furthermore, as a logical consequence, we'd be unable to define the point at which the singularity may occur at all, or even recognize it when it happens.
However, in order for the singularity to take place, AGI needs to be developed, and whether that is possible is quite a hot debate right now. Moreover, an algorithm that creates superhuman intelligence (or superintelligence) out of bits and bytes would have to be designed. By definition, a human programmer wouldn't be able to do such a thing, as his/her brain would need to be able to comprehend concepts beyond its range. There is also the argument that an intelligence explosion (the mechanism by which a technological singularity would theoretically be formed) would be impossible due to the difficulty of the design challenge of making itself more intelligent, getting larger proportionally to its intelligence, and that the difficulty of the design itself may overtake the intelligence required to solve the said challenge.
Also, there are related theories involving machines taking over humankind and all of that sci-fi narrative. However, that's unlikely to happen, if Asimov's laws are followed appropriately. Even if Asimov's laws were not enough, a series of constraints would still be necessary in order to avoid the misuse of AGI by misintentioned individuals, and Asimov's laws are the nearest we have to that.